Simulation and Verification of Hybrid Systems Based on Interval Analysis and Constraint Programming

Hybrid systems are systems consisting of discrete changes and continuous changes over time. Various systems in which computers reliably interact with their physical environment are modeled as hybrid systems. Simulation and verification of hybrid systems are done by integrating the computation of continuous dynamics and discrete changes, and by handling the uncertainties and computation errors. However, the computation of hybrid systems is often difficult, and may produce qualitatively wrong results, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations. This thesis is intended to provide a framework for nonlinear hybrid systems based on interval analysis and constraint programming. The detection of discrete changes in hybrid systems plays a significant role in the simulation and verification. We formulate the problem as a hybrid constraint system (HCS), which consists of instantaneous constraints, continuous constraints on trajectories (i.e., continuous functions over time) and guard constraints on states causing discrete changes. We implement a technique for solving HCSs by coordinating (i) an interval-based solving technique for nonlinear ODEs, and (ii) a constraint programming technique that reduces the interval enclosures of solutions. The technique generates a set of boxes smaller than a specified size that enclose the theoretical solution. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures while guaranteeing that the enclosure contains a solution, and it reliably solves HCSs with nonlinear constraints. Next, we present a bounded model checking method for hybrid systems. It translates a reachability problem of a nonlinear hybrid system into a predicate logic formula involving arithmetic constraints, and checks the satisfiability of the formula based on the satisfiability modulo theories (SMT) method. We tightly integrate (i) an incremental propositional satisfiability (SAT) solver to enumerate possible sets of constraints and (ii) an interval-based solver for HCSs to solve the constraints described in the formulas. The HCS solver verifies the occurrence of a discrete change by computing a set of boxes that enclose continuous states that may cause a discrete change. We exploit the existence property of a unique solution in the boxes computed by the HCS solver as (i) a proof of the reachability of a model, and (ii) a guide in the over-approximation refinement procedure. Our implementation, called hydlogic, successfully handles several examples including those with nonlinear constraints.